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# AMATH 342  Assignment 2 solution

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2. (a) (5 points) Using Theorem 2.1, derive the three-step Adams-Moulton method.
(b) (5 points) Using Theorem 2.1, derive the three-step Adams-Bashforth method.
3. (a) (5 points) Using Theorem 2.1, derive the two-step Nystrom method.
(b) (5 points) Implement the 2-step Nystrom method to solve the following scalar ODE:
y
0 + y = sin(t
2
), t ≥ 0, y(0) = 0.
Plot y as a function of t for t ∈ [0, 8]. Take h = 0.04.
Regarding the implementation, use Euler’s method to find y1. Use the Nystrom method
for all following time steps.
4. (5 points) Exercise 2.6 of book.
5. (a) (5 points) Construct the Gaussian quadrature formulae for the weight function ω(t) ≡
1, 0 ≤ t ≤ 1, of order two.
(b) (5 points) Construct the Gaussian quadrature formulae for the weight function ω(t) ≡
1, 0 ≤ t ≤ 1, of order four.
(over)
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6. Consider the linear ODE system
y
0 = Ay where A =

−20 10 0 0
10 −20 10 0
0 10 −20 10
0 0 10 −20

. (1)
Let the initial condition be given by y(0) = [1 1 1 1]T
.
(a) (5 points) Write down the 3-step Adams-Bashforth method for the given linear ODE.
Implement this method to solve (1) over the time interval t ∈ [0, 10]. At each time step
compute the Euclidean norm of the solution and plot this Euclidean norm as a function
of time. In your implementation, define h = 10/N, with N a positive integer. Given
that the norm of the true solution approaches zero as t → ∞, find out for which values
of N ≥ 50 the numerical method gives you a good approximation.
Regarding the implementation of the 3-step Adams-Bashforth method, use Euler’s method
to find y1 and y2. Use the 3-step Adams-Bashforth method for all following time steps.
(b) (3 points) Repeat part (a) but use a 2-step BDF method instead of the 3-step AdamsBashforth method.
Regarding the implementation of the 2-step BDF method, use the Backward Euler
method to find y1. Use the 2-step BDF method for all following time steps.
(c) (1 point) According to you, which method is best for this problem, the 3-step AdamsBashforth method or the 2-step BDF method? Explain your answer.
7. (5 points) The R¨ossler attractor arose from studying oscillations in chemical reactions. The
equations describing this system are given by
dx
dt = − (y + z)
dy
dt =x + Ay
dz
dt =B + xz − Cz
where A, B and C are system parameters. Take A = 0.2, B = 0.2 and C = 5.7. Implement a
3-stage ERK method of order 3 to solve the R¨ossler system of equations (you can choose any
3-stage ERK of order 3 that you want, but write down which one you chose). Let t ∈ [0, 500]
and take h = 0.02. As initial condition take (x0, y0, z0) = (0, 0, 0). Plot a 3D figure (with xyz
axes) of the solution.
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